Sour, I think you pretty much covered it. Rabbits kept on the ground are exposed to any diseases or parasites the wild rabbit population may have, and will probably get them. The frustrating thing about diarrhea in rabbits is that by the time you notice the symptoms, even if you identify the cause correctly and start the appropriate course of treatment, the animal dies anyway (often within 24 hours).
The usual treatment for cocci is usually Corid or Sulmet, or a veterinarian may have another drug they prefer. Treatment usually needs to be done for several days, then repeated a couple of weeks later.
Another possible cause of diarrhea in rabbits is a condition called enterotoxemia. As I understand it, it's basically an overgrowth of certain types of bacteria, fueled by too much carbohydrate in the diet. We don't usually think of grass as being high in carbohydrates, but it can be, particularly if it is experiencing rapid growth. Hay is usually less so (because the grass is typically in a more mature stage when cut). The treatment for this type of diarrhea is a high-fiber, low carb diet (which may mean just hay and water, nothing else), possibly antibiotics, and doses of beneficial bacteria (a product like Bene Bac). I have found plain, dry, rolled oats to be helpful at firming up simple cases of loose stools; offering a tablespoonful or so might be a good idea.